Integral of x^i

Learn how the complex integral of x^i can help us find the integral of sin(ln(x)) and cos(ln(x)) without using integration by parts. Complexifying the integral is an integration technique that is usually not taught in calculus 2.
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Check out my 100 integrals for more integration practice for your Calculus 1 or Calculus 2 class. • 100 integrals (world r...
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[@gcewing]

"Killing two birds with one stone" is a strangely appropriate metaphor, seeing as "calculus" is the Latin word for "stone". :-)

[@bishalgupta49]

@VeryEvilPettingZoo 😂

[@Qoow8e1deDgikQ9m3ZG]

you are wrong, it is killing two eggs with one nail

[@umbratilecandelaio7830]

@Bob Trenwith calculus means simply small stone. Then by extension it was used for those pebbles used for counting.

[@umbratilecandelaio7830]

​@Bob Trenwith From Charlton T. Lewis, Charles Short, A Latin Dictionary: calcŭlus , i, m. dim. 2. calx; cf. Paul. ex Fest. p. 46. I. In gen., a small stone, a pebble.

[@turbotaleggio8425]

In modern italian “calcoli” means both calculations and kidney stones

[@OLApplin]

6:55 "I don't like to be on the bottom, I shall be on the top "... your choice man!

[@TaygaHoshi]

Olivier L. Applin isn't it?

[@raghavgovind2580]

Smash Boy He meant something else

[@morjithmattapalli9531]

It's his choice (most indians will get this)

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This channel is awesome *isn't it?*

[@blackpenredpen]

Suave Atore thanks!

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Yes it is

[@algobrax]

i(sn't + t)

[@gautamchopra9939]

Ofcourse.

[@andrzejsadownik4947]

@my pp is hot it(derivative of ns + 1)

[@lukevaneyk3896]

I feel that teaching with colors as you do aids a lot in the grasping of these problems. When reading textbooks, it isn't always clear how they get from one step to the next. Your demonstrations make it much clearer.

[@blackpenredpen]

Luke van Eyk thank you!!! And Oreo made it cuter too!!!

[@calyodelphi124]

Y E S Yankee Echo Sierra Y E S The colors help *so much* and I love BPRP for it.

[@jorenheit]

I started doing this in class after I'd been watching his videos. It really adds to the clarity! My students seem to appreciate it at least :-)

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Joren Heit you just have to master the single-handed marker swap technique!

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I just found out why it is called black pen red pen

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This is pure joy.

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kujmous

[@verainsardana]

10:35 "if you are obsessed with +c" i laughed so hard

[@user-en5vj6vr2u]

Yeah lmao nobody likes c

[@AndDiracisHisProphet]

and some people say imaginary numbers aren't useful

[@passecompose7484]

AndDiracisHisProphet Actually wireless transmition can't be explained without system of complex numbers

[@AndDiracisHisProphet]

Yes, of course it could. It is just more convenient to do it this way.

[@calyodelphi124]

It could (with an asterisk), but not as concisely and as accurately as you can do with complex numbers. When dealing with any electronics that uses an AC signal (mains power, audio electronics, RF electronics, etc.) you're no longer dealing with simple real quantities like resistance, capacitance, and inductance. You're now also dealing with complex quantities like capacitive reactance and inductive reactance, which are the imaginary parts that combine with the real part resistance to form the complex value of inductance. In the face of an AC signal, capacitors and inductors exhibit a form of resistance to the time-variant signal. But it's not real resistance. It's imaginary resistance--i.e. reactance. They also induce phase shifts to the AC signal as well. Electrical engineering very, very quickly dives deeep down the calculus rabbit hole when you venture into the AC world. In the DC world, 90% of the math you do boils down to basic arithmetic. And there's still plenty of that in the AC world, too! But there's also calculus in the AC world. Lots of it. And it's all math that is presently above my head, but I'm still doing what I can to suss it out in my own free time in lieu of being able to afford to go to university. :) (Heck, I'll probably have a better, more intuitive understanding of it all than what any university course could probably teach me. Leg up for when I go back to school!)

[@calyodelphi124]

How would you describe the vectors purely numerically? You could probably use the linear algebra approach of 2x1 matrices, but then you have to remember all the additional mathematics that comes with linear algebra as well. The alternative approach is to describe those vectors numerically using complex numbers. Or if you need a notation that describes the vector's length and angle, you'll need z e^(i theta), and you'll need to know how to convert between the two. Remembering that z e^(i theta) = z(cos(theta) + i sin(theta)) is a lot easier than having to remember a whole bunch of extra stuff regarding linear algebra and matrix algebra. :P

[@passecompose7484]

Bob Trenwith It seems possible, but very inconvenient

[@adb012]

The title of this video should not be "integral of x^i" but "how to solve integral of sin(ln x) and of cos (ln x) at the same time and much easier than using substitution and integral by parts"

[@mike4ty4]

One last thing though - technically, you would need to prove the power rule in fact is valid for complex numbers to make this airtight, though I suppose you're going on that that's already been done: nonetheless, it wouldn't necessarily be justifiable at, say, the usual Calc II-like level where this seems to target, to do that. But that is easily remedied: you can first assume _as a heuristic or hypothesis_ that the power rule will work for n = i (seems reasonable, no?) and then after carrying through with it, go back and _differentiate your final answer_ to see that you do indeed get cos(ln(x)) + i sin(ln(x)), thereby proving that not only does the rule apply for that imaginary power but also that you have indeed integrated the two integrals that you wanted to "with one stone". This is not circular because you did not reference the conclusion as justification; rather simply only took it as a hypothesis to then later be verified.

[@MattMcIrvin]

I think you could prove it using Cauchy's residue theorem, but that's college complex analysis stuff... very beautiful though, because it makes a lot of tough integrals melt away magically.

[@CatchyCauchy]

Just do it with implicit differentiation. Let y=x^n, where n is element of all complex numbers. y=x^n |take the ln ln(y)=ln(x^n) = n ln(x) |differentiate with respect to x (1÷y)×y'= n÷x |multiply both sides by y y' = ny÷x | y= x^n y'=nx^n÷x =n×x^n-1 That's how you derive the power rule, for all integers, and if you integrate you just do the reverse

[@tensor131]

@@CatchyCauchy I think it's bit heavier than that. Problem is that all the standard calculus results (real) relate to gradients and area .. and that is how the fundamental theorem of calculus (which is where the n+1 rule emerges) comes from. Now as soon as you make the function non-real all those concepts disappear so I guess everything needs to be re-derived? Fortunately most of the standard results still hold in complex analysis ... and you get a whole lot of (beautiful) extra embellishments :)

[@CODPIMPS4210]

I love your channel. I'm currently in high school, working on my second year of calculus, and I love challenging myself with your higher level content, and your explanations are easy to follow and in depth enough, but not too in depth to the point of being boring. Keep up the great videos.

[@disasterarea9341]

this is the most wholesome channel on youtube

[@whozz]

I recommend you to see the graph of the integral of sin(ln(x)). It is really cool, the graph repeats itself in ever larger scales as x grows to infinite.

[@gnikola2013]

WHoZ Wow, it's looks like a kind of fractal

[@whozz]

Tecnically it's not a fractal. The right name to this property is "self-similarity". In that the specific function the property works as following: f(xe^(2pi)) = e^(2pi)*f(x)

[@whozz]

Kiritsu The first time I saw this function I also thought it was "kind of fractal" hahahaha math.stackexchange.com/questions/2407743/is-the-function-fracx2-cdot-sin-lnx-cos-lnx-a-fractal

[@evilotis01]

i love how this dude sits and works out crazy-ass maths problems on his own at home, and when he comes up w something great, he shares it w everyone on KZbin. his excitement is infectious, and i love watching his channel

[@user-co6rg9jt9x]

I love how happy you got after seeing this patern. I've recently stopped focusing on mathematics, because of my software engineer carrier, but that video reminded me the way I felt after solving things. Thank you for the video.

"Tonight I figured this out" omg dude, you nights must be pretty intense.

[@toastybread7375]

Can we take a second to appreciate the pen lid removal at 6:16

[@blackpenredpen]

ToastyBread great catch!

[@roberthayter157]

Your enthusiasm is infectious. I can feel the joy you felt when you worked this out.

[@LeeSawyer]

I've become addicted to this guy's videos!

[@alexturk6389]

love how he switches so seamlessly between marker colors

[@narusasu7279]

Absolutely brilliant! Your cheerful attitude plus the way you make a mind-blowing problem sound so simple is what makes me want to watch MORE of your videos! So good!

[@Alians0108]

No matter what method you use, maths stays consistent once again. Never fails to blow my mind.

[@theMathSinger]

Just now seeing this. Very, very clever! Well done, sir.

[@blackpenredpen]

Thank you! : )

I'm more excited about your excitement than the solution itself. :D But I understand the genuise way (I could never figure out by myself). Really good job! Booth thumbs up.

[@blackpenredpen]

Daniel Gschösser thanks!!!! I am very glad to hear!!

I've to say thanks. Your work for society in science is unpayable. Cheers man!

[@PuzzleQodec]

I'm so curious now about what other nuggets of curious investigation are yet to come. Can't wait for the next video!

[@strafeae4618]

One of my favorite things about this channel is that you respond to EVERY SINGLE COMMENT. You really do make KZbin a greater place not only because of your mathematical genius, but because you are just a wonderful person

[@blackpenredpen]

Hi Strafe, thanks for your awesome comment! I do try to respond to as many comments as possible. I also feel sorry that when I can't respond to everyone since I have been getting LOTS of comments per day. I love them all, but I just can't get to every single one of them. Thank you again. Reading comments always make my days brighter!

[@spelunkerd]

Your enthusiasm is infectious, thanks!

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This is without doubt one of the coolest methods I have ever seen.

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That's pretty delightful!

[@Joslinmeister]

Love the videos, man. Thanks!

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Love your videos so much :D

[@terapode]

Your videos are ALWAYS GREAT!! I like the way you think to solve the problems.

[@blackpenredpen]

thank you!!!!!!!!!!!!

[@chasemarangu]

"so good!" is now officially a blackpenredpen meme.

[@Exxag]

I have just randomly stumbled upon your channel and I have to say that I really enjoy your passion. Only the sound quality suffers a bit sometimes, but still awesome and interesting videos!

[@DoReMeDesign]

I loved this so much, your enthusiasm and the maths that I was able to follow, thank you.

[@calyodelphi124]

Your unbridled eagerness to share your epiphany far and wide here on youtube speaks to a passion for teaching that I'm sure would make my late mom proud. Also I just get so squeeful watching geeky people like me get giddy and show off the geeky things they do for fun.

[@johnchristian5027]

love your vids, keep up the good work !

[@alkhuarizmiacademy2375]

Excellent way of Teaching Mathematics.... Keep it up! I like your combination of Black and Red ❤️

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I love your enthusiasm

[@rafael7696]

I am Spanish. I love this channel. THANKS and more videos!!

[@nicktarantino7188]

He seemed so excited at the end this shit is great.

[@MaxxTosh]

Can you do a series of videos for integral, derivative, etc, of the hyperbolic functions and their relationship with i? Or, if possible, do a video with quarternions?

[@paoloazzini9003]

I clicked the video just to see the domain of integration, because i wanted to try to solve it trought complex integration method. I didn't expect you would calculate the primitive. Nice!

[@jheeze3816]

My man really out here teaching calculus in supreme, respect

[@NilsThylen]

You are awesome and your wonderful attitude made me laugh. Fantastic!

[@mike4ty4]

@486 s (8:08) - CHEERS for actually doing it the way that makes clear the underlying math rules being employed and not just that hackneyed old "FOIL it out" stuff.

[@gavinridley5727]

Thanks for the great vid man! :D

[@michaelroditis1952]

I love your passion

[@matthewfrench428]

Thanks for sharing this with us!!

[@arkabarman8388]

I think you are a genius. The way of your teaching is fabulous

[@ytrichardsenior]

Once, I wanted to 'spin' a telescope mirror with epoxy resin. But I needed to calculate the focal length of the mirror based upon something I could measure, like the change of 'height' of the edge of the liquid as it rose up the container due to centripital force. I could place a marker on the edge of the circular container and spin the liquid until the edge reached the marker.. But how high above the flat resting level of the liquid to place the marker? After an hour or two and a little bit of differentiation by rule, I solve the problem. I think I was even happier than this guy is here :) One of the greatest achievements of my mathematical life (I'm an engineer really, not a mathematician). Of course subsequently I lost the piece of paper on which I did the work, and have never been able to recreate it :)

[@quadruplay9788]

You know shit gets real, if blackpenredpen takes out a blue pen

[@sruelle1]

That's a great video, it helps me a lot with my calculus teaching

[@joshwriteman3188]

Your videos gave me the opportiunity to show my class in Israel the beauty of Mathematics.Thank you!

[@blackpenredpen]

josh writeman what a wonderful comment! Thank you for making my day (Friday! Woohooo!)

[@Zyx3ds18]

My man's is out here wearing a supreme jacket. Damn homie I can't believe you been flexing on us like that since you got big

[@justdad3681]

Love ur passion

[@dospaquetes]

This is awesome. Can't wait to show it to my students!

[@charredmars2659]

I integrated by Parts and got that but I’m kind of sad now because the way you did it is really cool and not boring IBP. Haha great video

[@donmoore7785]

Always interesting content!

[@tesfayebabore6862]

You are artist of maths! what a talent!

[@TheMauror22]

So great! This was crazy!

[@42tai-chan]

math always seems to amaze me

[@yerr234]

(never ceases to )*

[@JSSTyger]

You can tell that red pen and black pen are friends. They finish eachother's equations.

[@dipesh-singla]

Loved your channel. From which country you are from? Which class syllabus is these types of questions in your country

[@RameshChandra-uh5rw]

Now my phobia of maths is getting converted into love for it... All thanks to you.

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Ay he's rocking a supreme jacket 🔥💯👌

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Isn't it

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My dawg fresh af

[@andrewstallard6927]

Another mind blowing video. I teach calculus myself right now and I did not anticipate that equating of coefficients using real and imaginary parts in the end. When my students are ready (They are still working on limits) I will certainly introduce them to your channel.

[@blackpenredpen]

Andrew Stallard thank you!! I should be doing a better job in terms of organizing the topics that I am doing. Right now, I am just doing videos on whatever it's in my mind. Lol

[@blackpenredpen]

Andrew Stallard oh, I do have some videos on limit already, plus my site www.blackpenredpen.com

[@BlackFiresong]

+blackpenredpen Perhaps in time, you can make playlists by topic or theme? :)

[@blackpenredpen]

BlackFiresong I have the playlists (for HW sol for my students) on my calc resource site. But for my YT channel now... it's very unorganized haha. I guess it's just a math person's habit...

[@JithendraJithendraReddy]

Super sir Your teaching is very understandable and simple

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Where did you buy your jacket? I love it.

[@udit410]

found this channel later but worth it

[@digbycrankshaft7572]

Love your work man. Wish you had been my maths teacher. 👍

[@duncanw9901]

The fact that that was your own making earned a sub

[@zamknijryj1780]

Hi. I have a question about second integral. We have i in front of integration sign. What happened with this after integration sin(ln x)?

[@MVEC_MuthuKumar]

Doubt!, Is it possible to do that without multiply or dividing i+1 but how can satisfies x^i+1 @ 4:00 or i didn't understand that part only 4:00 Thankyou

[@StarEclipse506]

never smashed the mfking like button so fast

[@zombiesalad2722]

Never thought of solving it that way lol. Awesome video

[@Dr-Tehnix]

I like you, keep it up!

[@JeaneAdix]

Hey bprp I wonder if you can do some more proofs videos, I like your insights, thanks!

[@blackpenredpen]

John Ecstasy I have induction videos on my to-do list.

[@inesantoniosanchezgutierre664]

Exellent, so simple, isn´t it. A few years ago I watched a video where a group of men honored a salsa singer. They shouted at the end : "The world needs salsa". Let me tell you man, you really love maths. I´d love my students and colleagues to be like you. What we could shout as one has to be: "The world needs maths". It really needs!! Above all those countries like mine the ones that nowadays continue to worshiping politicians. Go ahead guy!!! You have too much to give yet. Congratulations.

[@aurelstrat1829]

EXCELLENT PRESENTATION! THANK YOU, SIR!

[@DavyCDiamondback]

I never took complex analysis (looking into the Reimann Hypothesis), and tried to do this myself another way, and am sure I messed up with all the substitutions when I wound up with something that looked more like a derivative. It's nice come back to this video and remind myself why integrals and derivatives of complex powers behave so nicely

[@dr.rahulgupta7573]

Excellent presentation ! Vow !!

[@chuckstarwar7890]

Great talk. Most books on complex functions discuss these tricks.

[@mieumieu8417]

Your channel is good for me because I can learn math and English at the same time. But I'm not good at either yet.

[@fvveb2141]

I love this channel

[@blackpenredpen]

Eva Martane thank you!!!

[@Shadowsphere1]

I already knew Euler's formula from differential equation use, but I never would have thought of using it like that before now...

[@ricardinhovorkes4876]

So cool thanks again man 👍

[@golammartuzahossain6748]

I have a question.How do we define complex exponential with a negative real number on base.Like (-2)^i??Certainly the one in the video doesn't suffice because the natural logarithmic function would be undefined then. :/

[@user-fy5tn7sy3t]

You have pretty ways for math so, Ilearned you more ... Thanks teacher but I have question Could u help me to get series of sinx/x ?

[@aidan8858]

Why do we use e and the natural log to cancel each other out? Why not like, 4 and log base 4 for example? Or just any other number. Wouldn't that work too?

[@whozz]

When we evalute a definite integral of x^i, what we are calculating? Volume, area or something else?

[@dyer308]

Yayy another blackpenredpen complex number video

[@blackpenredpen]

Yay!!

[@chuckjls]

Those crazy functions!

[@evgeniivol8369]

So good!

[@epiccheeseburgercat]

Wish my profs had this much enthusiasm when teaching.

[@igoralvespratescostaleite3577]

Tanks for the short Oreo's video. He should have his own channel.

[@vedant3148]

I like Your Teaching Technique

[@richardfredlund3802]

This video is awesome :)

[@deepak-ds6iw]

I am from India .we use that but your channel is good because you pickup some uniq problem of math.

[@srpenguinbr]

Do a video solving a simple double integral, that would be nice